The author studies a classical mechanical system of weakly coupled rotators on a one-dimensional lattice. This paper is very close related to certain numerical investigations on a chain of weakly coupled rotators [\textit{G. Benettin}, \textit{L. Galgani}, and \textit{A. Giorgilli}, Phys. Rev. A 13, 1921 ff. (1976); Theor. Math. Phys. 29, 1022 ff. (1976); I1 Nuovo Cimento 52 B, 1 ff. (1979); and some preprints]. One gives an explanation of these investigations. It is proved that for any site in the system there is a ''large'' set of initial conditions for which there exists a canonical change of variables such that the trajectory of that site in the transformed system is essentially indistinguishable from that of an integrable system for a long (but finite) time. Alternatively, the trajectory of this site lies very close to torus in the phase space of the system for times very long in comparison with the typical period of the unperturbed rotators. All the estimates in this theory are independent of the number of degrees of freedom in the system.